Communications in Mathematical Physics

, Volume 317, Issue 1, pp 157–203 | Cite as

Cyclic Cocycles on Twisted Convolution Algebras

Article

Abstract

We give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. For proper étale groupoids, Tu and Xu (Adv Math 207(2):455–483, 2006) provide a map between the periodic cyclic cohomology of a gerbe-twisted convolution algebra and twisted cohomology groups which is similar to the construction of Mathai and Stevenson (Adv Math 200(2):303–335, 2006). When the groupoid is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial techniques to construct a simplicial curvature 3-form representing the class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial curvature 3-form to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bott R.: On characteristic classes in the framework of Gelfand-Fuchs cohomology. Astérisque 32–33, 113–139 (1976)MathSciNetGoogle Scholar
  2. 2.
    Bott R., Tu, L.: Differential Forms in Algebraic Topology. New York: Springer, 1982Google Scholar
  3. 3.
    Bouwknegt P., Carey A.L., Mathai V., Murray M.K., Stevenson D.: Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys. 228(1), 17–49 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    Bressler, P., Gorokhovsky, A., Nest, R., Tsygan, B.: Deformations of gerbes on smooth manifolds. In: K-Theory and Noncommutative Geometry, EMS Series of Congress Reports, zürich: European Mathematical Society, 2008, pp. 349–393Google Scholar
  5. 5.
    Brylinski, J.-L.: Loop Spaces, Characteristic Classes, and Geometric Quantization. Birkhäuser, Boston-Basel-Berlin, 1993Google Scholar
  6. 6.
    Carey A.L., Mickellson J., Murray M.K.: Bundle gerbes applied to quantum field theory. Rev. Math. Phys. 12(1), 65–90 (2000)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Connes, A.: Noncommutative Geometry. London New York: Academic Press, 1994Google Scholar
  8. 8.
    Crainic, M.: Cyclic Cohomology of Étale Groupoids; The General Case. K-Theory, 17 (1999)Google Scholar
  9. 9.
    Dupont, J.L.: Curvature and characteristic classes. Number 640 in Lecture Notes in Mathematics. Berlin-Heidelberg-New York: Springer-Verlag, 1978Google Scholar
  10. 10.
    Felisatti M., Neumann F.: Secondary theories for simplicial manifolds and classifying spaces. Geometry & Topology Monographs 11, 33–58 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gawe¸dzki, K., Reis, N.: WZW branes and gerbes. Rev. Math. Phys. 14(12), 1281–1334(2002)Google Scholar
  12. 12.
    Giraud, J.: Cohomologie non abélliene. Number 179 in Die Grundlehren der mathematischen Wissenschaften. Berlin-New York: Springer-Verlag, 1971Google Scholar
  13. 13.
    Gorokhovsky A.: Characters of cycles, equivariant characteristic classes, and Fredholm modules. Commun. Math. Phys. 208(1), 1–23 (1999)MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Hitchin, N.: Lectures on special Lagrangian submanifolds. In: Vafa, C., Yau, S.-T., eds., Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds, Number 23 in Studies in Advanced Mathematics, Providence, RI: Amer. Math. Soc., International Pres, 2001, pp. 151–182Google Scholar
  15. 15.
    Arthur J., Andrzej L., Konrad O.: Quantum K-theory, I. Chern character. Commun. Math. Phys. 118(1), 1–14 (1988)MATHGoogle Scholar
  16. 16.
    Loday, J.-L.: Cyclic Homology. Number 301 in Die Grundlehren der mathematischen Wissenschaften, 2nd ed. Berlin-Heidelberg-New York: Springer-Verlag, 1998Google Scholar
  17. 17.
    Mathai V., Stevenson D.: On a generalised Connes-Hochschild-Kostant-Rosenberg theorem. Adv. Math. 200(2), 303–335 (2006)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Mathai V., Stevenson D.: Entire cyclic homology of stable continuous trace algebras. Bull. London Math. Soc. 39(1), 71–75 (2007)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Moerdijk, I.: Orbifolds as Groupoids: an Introduction. In: Adem, A., Morava, J., Ruan, Y. eds., Orbifolds in Mathematics and Physics, Number 310 in Contemporary Mathematics, Providence, RI: Amer. Math. Soc., 2002, pp. 205–222Google Scholar
  20. 20.
    Moerdijk I., Mrčun J.: Introduction to Foliations and Lie Groupoids. Cambridge: Cambridge Univesity Press, 2003Google Scholar
  21. 21.
    Quillen D.: Algebra cochains and cyclic cohomology. Publ. Math. IHES 68, 139–174 (1989)Google Scholar
  22. 22.
    Tu J.-L., Xu P.: Chern character for twisted K-theory of orbifolds. Adv. in Math. 207(2), 455–483 (2006)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ColoradoColoradoUSA

Personalised recommendations